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RE: The Beauty and Mystery of Mathematics
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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 RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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Another look at Platonism. We communicate via shared concepts. If we didn't have shared concepts, how could we understand one another? But concepts can be fuzzy around the edges and that fuzziness can get us into trouble. The classical Greeks, while guilty of introducing the Platonic forms, also came up with a workaround for the fuzziness of concepts: the axiomatic method.
"Color" is a fuzzy concept. More specifically, the color "blue" is a fuzzy concept. I sat on the balcony of a fancy hotel in Nusa Dua on Bali, enjoying the tropical breeze and a second cup of coffee after a late breakfast. My companion was a pretty young Javanese girl. (Don't miss your cue here, Vega. Doesn't this make you thirsty?) She was teaching me bahasa indonesia the Malay dialect that is the official language of polyglot Indonesia. Pointing to the border of the menu cover she pronounced hijak. Hijak is the color of green beans in Malay, but the menu border had a decided bluish cast. Noting the blue-green color of the band bordering the sleeve of my polo shirt, I pointed to it and responded, "Hijak."
"Tidak! Tidak!" she objected. "Biru." Blue, not blue-green. She held the menu up to my shirt, "Can't you tell the difference?" Indeed the shirt sleeve was that tiny bit more blue.
"Yes, but to me they are the same color. I can see the difference now, but if you had not pointed it out, and you asked me at dinner, I would say they were the same color." Then I started laughing.
"What's funny?"
"The same thing happened in the third grade, only in reverse." The teacher was a red-haired gringa with the build and manner of a Marine drill sergeant. In Spanish class she held up a card and asked what color it was. No one answered. People either didn't know, or some of the quicker Spanish speaking members of the class foresaw a difficulty I did not. I raised my hand and said, "Alazán."
"No," said the teacher, "marrón."
"But, Mrs. DeV., marrón is the color of chestnuts. Alazán is the color of a sorrel horse, like the card." On the ranch where I spent every summer from the age of four, these were important distinctions. Some of the horses had names, but most of the steeds in the remuda were described simply by their physical characteristics, "The big chestnut mare, the piebald sorrel gelding…" And 21 of the 22 families who lived on the ranch spoke Spanish at home.
I persisted. Mrs. DeV. assigned me extra work for defying the teacher. Fuzzy concepts can get you in trouble.
The axiomatic workaround is this. You set out a certain set of statements about a concept you wish to discuss like "line" or "point". These statements are called "axioms". Everyone agrees that everything said in the future about the concept will be deduced from the axioms by certain rules of logic. This eliminates large areas of disagreement. There is at present still a bit of discussion about the rules of logic to be employed, but most mathematical logic agrees with ordinary patterns of thought.
This approach has one of the characteristics of Platonism. The mathematical objects are accessible only to thought. Indeed they are thoughts. What else could a concept be? Concepts exist, else how could we communicate? But they do not transcend the human mind. They are the shared products of human minds.
Concepts can be taught. They can be invented. Invented concepts can be consistent with other, previously tested concepts, or they can be wrong. Useful, striking, consistent concepts are often said to be "discovered." Indeed, that's just how it feels sometimes.
Concepts have fallen into and out of favor over the centuries. For example Leibnitz, in his development of calculus, spoke of infinitesimals: numbers smaller in magnitude than any positive number, yet still not zero. Newton did too, at first, but later abandoned infinitesimals. The trouble with infinitesimals is that they violate the rules or ordinary arithmetic. In ordinary arithmetic no matter how small a number may be, if it isn't zero, there's another number, still smaller, between it and zero. In arithmetic there are no nonzero numbers smaller than all the others.
Infinitesimals were ridiculed in such a withering and effective manner by Cardinal Newman ("ghosts of departed quantities") that mathematicians of the generations after Newton and Leibnitz abandoned them. Finally in the 19th century Cauchy and others described the idea of "limit" in sufficiently clear and logical terms that calculus was declared to be a logical pursuit after all.
When I took calculus in 1956, infinitesimals were still being ridiculed, then by mathematicians, not clergymen. But they still appeared in profusion in physics texts and the lower orders of calculus texts.
Then along came Abraham Robinson, the inventor of Non-Standard Analysis. Using sophisticated topological and logical concepts, he extended the idea of the set of real numbers and the rules for manipulating them, to include things that behaved just like Leibnitz's infinitesimals, if you used the same symbols for the new operations as the ones used for ordinary arithmetic. I'm pretty sure Leibnitz's ideas of "infinitesimal" were quite different from Robinson's, yet….who "discovered" infinitesimals?
What does "existence" mean to mathematicians? Most would evade the question. Euclid said you could draw a line between any two points. Hilbert in his more rigorous presentation, 2,400 years later, said, "If A and B are two points, there exists a line containing them." This is a statement about conceptual existence. The concept "line" has the property ascribed to it by Hilbert's axiom. It doesn't mean that any physical objects have the properties mentioned, nor that something exists in never-never land. It means that the concepts "line" and "point" abstracted from our physical experiences can safely be said to behave this way.
This paper from 1962
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1359523/pdf/jphysiol01247-0121.pdf
cited at least 8700 times, indicates that there are specific brain responses to things we would call lines and points in the visual field, at least for cats. I don't know whether anyone has wired up humans to test us for these responses.
This suggests that the concepts "line" and "point" may describe fundamental operations of our brains, honed by evolution over the ages to deal with our environment. Who wouldn't be excited to discover new relations between such fundamental brain responses? The responses (not the concepts, as far as we know) are indeed likely to transcend the human mind, extending to cats, dogs, porcupines….
Ernest Thompson Seton argued that crows could count to at least ten…
And the last words of the famous African Grey Parrot Alex, before he died unexpectedly in the night from unknown causes, were his usual, "You be good. See you tomorrow. I love you."
http://en.wikipedia.org/wiki/Alex_(parrot)
RNJ
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Date Nov. 20 2013 21:40:02
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to guitarbuddha)
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quote:
Bill this statement quote: Nevertheless, I think mathematical structures transcend, and are independent of, the human mind (forget Platonic Forms). When mathematicians make a "discovery," or solve a problem (Fermat's Last Theorem), they are bringing forth something that existed prior to their "discovery"; something that was just waiting for a human mind (or minds) to reach the level of sophistication required to "crack the code," so to speak.
And this one quote: In fact, one can say that religion is culture. And it pervades their entire existence in everything from their overall belief system to the laws that govern...Have an awful lot in common. If one simply replaces maths with god in the first the uncomfortable equivelancy is painfully clear.
But I did not intend the first statement to refer to God at all, GuitarBuddha. When I refer to something existing prior to a human mind discovering it, I am referring to what might be called certain "symmetries" that exist (almost as a natural phenomenon, for lack of a better explanation), independent of the human mind, and certainly independent of anyone's concept of a god or gods. I place my faith in science, not religion. I do think that certain "symmetries" (again, I don't know any other way to describe them) such as mathematical structures exist, or are embedded, in the natural world, and we "discover" them when a certain level of mental sophistication enables us to perceive them. A sort of Eureka! moment.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 22:06:40
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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Bill-
May I suggest Carl B. Boyer's "History of Calculus and its Conceptual Development"? It details the long struggle to formulate differential and integral calculus in a logically consistent way. It contains a handful of clinkers, since Boyer, writing in the 1930s was not up on the latest stuff even for the 1930s, but the clinkers are minor, not affecting the overall story. It's an interesting history that treats the "discovery" of calculus as an intellectual adventure in the development of concepts.
http://www.amazon.com/History-Calculus-Conceptual-Development-Mathematics/dp/0486605094
I mentioned Morris Kline's "Mathematics, the Loss of Certainty" previously. It portrays the 20th century debates on the foundations of mathematics from a mathematician's perspective, not a philosopher's.
http://tinyurl.com/lfnymxz
Mathematicians and physicists are far from perfect. But having been trained as a mathematician and physicist, trying to read "philosophy of science" is more often than not frustrating to me. It seems to me there is often a mistaken emphasis, or they simply get things wrong.
It seems to bug Barry Mazur, too.
http://www.math.harvard.edu/~mazur/papers/plato4.pdf
I'm trying to read Teller's "An Interpretive Introduction to Particle Physics". He spends a whole chapter talking about "particles" as though they were some kind of physical reality and then gets the description of the Fock space of the Standard Model wrong, calling it a "tensor product of Hilbert spaces."
The developers of the misnamed "particle physics" just said, "Forget about 'particles' except to use it as shorthand for quantized wavelike perturbations of fields on Hilbert spaces."
At present it seems that the more accurate our physics models are, the less "physical" they become, in the sense of physical intuition. The physics models are constructed from abstract mathematical concepts. Trying to describe them in prose as philosophers often do, is doomed to failure.
RNJ
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Date Nov. 20 2013 23:02:21
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Richard Jernigan)
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Richard,
Having lived and worked in Indonesia for four years and Malaysia for four years (a total of eight years in the Malay Archipelago), I speak both Malay (Bahasa Melayu) and its Indonesian variant (Bahasa Indonesia) fluently. Not to be picky, but I think you meant "biru" (vice "baru") to mean the color blue. In both Malay and Indonesian "biru" means"blue," and "baru" means "new."
I do not agree that your example of "color" in Bahasa Indonesia illustrates "fuzziness." It appeared to be "fuzzy" to you because you did not grasp the subtlety of the language and applied your own understanding of color in English to the situation. This is often the case when learning a foreign language. A good example is the word "rice." In English, we refer to rice as "rice," whether it is growing in a rice paddy, in a bag at the supermarket, or on our dinner plate ready to eat. In Malay and Indonesian, however, there are four different terms for "rice," depending on its condition.
PADI refers to rice on the stalk growing in the field.
GABAH refers to unhusked rice separated from the stalk.
BERAS refers to uncooked rice on the supermarket shelf.
NASI refers to cooked rice ready to eat.
What we may sometimes think of as "fuzzy" is often just a linguistic misunderstanding. And it harks back to your statement about holding "shared concepts" with your interlocutor to prevent misunderstandings. That's the provenance of the old adage that to know a little of a foreign language can be dangerous. In the case of "rice" or "color" it is a benign misunderstanding, but one can get into areas where a little knowledge is more dangerous than no knowledge at all. In any case, in the example of "rice," Malay and Indonesian are anything but fuzzy. In fact, they exhibit greater precision than English.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 23:09:20
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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quote:
ORIGINAL: BarkellWH
What we may sometimes think of as "fuzzy" is often just a linguistic misunderstanding. And it harks back to your statement about holding "shared concepts" with your interlocutor to prevent misunderstandings. That's the provenance of the old adage that to know a little of a foreign language can be dangerous. In the case of "rice" or "color" it is a benign misunderstanding, but one can get into areas where a little knowledge is more dangerous than no knowledge at all. In any case, in the example of "rice," Malay and Indonesian are anything but fuzzy. In fact, they exhibit greater precision than English.
Cheers,
Bill
OK, I'll trot out one of R.L. Moore's favorite stunts. He would announce that in the work to follow, "point" and "region" were undefined terms, but with the proviso that they mean something. "You may not think that's a very strong requirement," he would say, "that a word should mean something."
He often sat in an oak straight backed chair behind an oak desk at the front of the room, dressed in a dark blue suit tailored to his athletic build, with an immaculately starched and pressed white shirt, conservative tie and handmade high top shoes of the softest calfskin. Both chair and desk were in good condition. The classroom was effectively reserved for Moore's use. It had beautiful slate blackboards, while other classrooms in the building had boards of painted Masonite. The rumor was that Moore had paid for the boards himself, so as not to burden the taxpayers with his preference.
Moore placed the palm of his hand on the desktop, and with a twinkle in his eye asked, "Mr. X, what is this."
"It's a desk, Dr. Moore."
Moore removed the drawers and stacked them in a corner. "Now what is it, Mr. X?"
"It's still a desk, Dr. Moore."
"If I had a saw and sawed off one corner," about four inches, he gestured, "what would it be then, Mr. X?"
"Still a desk, sir."
Moore proposed to saw off more and more of the desk, until Mr. X began to express some doubt about its integrity. Eventually Moore proposed to saw it right down the middle. "And now, Mr. X?"
"It's a desk sawn in two."
"Is that the same thing as a desk?"
"….not exactly."
"If I sawed a sixteenth of an inch to the right of the middle, would it still be a desk sawn in two?"
Mr. X abandoned ship at this point.
"Thank you, Mr. X for putting up with that." Addressing the class at large, "So you see, requiring that a word mean something may not be such a weak requirement."
Since all theorems were to be derived from Moore's axioms for topology (revolutionary at the time of their introduction) the opportunity was eliminated for fuzziness to cloud the discussion of "point", "region" and the many terms defined from them such as "continuum" , "arc", "simple closed curve", and so on for three years of study.
Thanks to Euclid--or whoever Euclid got the axiomatic idea from.
RNJ
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Date Nov. 21 2013 0:55:30
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Ruphus
Posts: 3782
Joined: Nov. 18 2010
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RE: The Beauty and Mystery of Mathem... (in reply to guitarbuddha)
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Well said; and I think BTW common sense is not aware that nature has no consciousness / aims.*
Yet, playing without aims has been found by biologists ( = it´s not my spontaneous assumpion).
Formerly, as you know, animals playing was assumed as always meaningful / task. ( Training skills, strenthening sociologically, etc.)
Over past years however observers discovered that advanced animals can play simply for the joy of it.
Maybe most popular example: Jack daws sliding down snowy slopes.
I would say, even as amateur observer you can see how they plain enjoy the slide without any competing with each other.
Intelligent species know the just-for-the-sake-of-it action.
Ruphus
PS:
* Ever thrilled by the fact that nature has no aim, I find it even more fascinating that it is sheer measure of time ( in our perceptive dimension) that enables incredible finesse to come about.
Time, the universal fertilizer, if you want.
-
... - And now, besides, it is five before twelve on earth.
Mere due to sapiens ape that prefers to dismiss beauty before the eyes and knit its own bovine entity of detouchment.
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Date Nov. 21 2013 19:15:29
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Ruphus
Posts: 3782
Joined: Nov. 18 2010
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RE: The Beauty and Mystery of Mathem... (in reply to guitarbuddha)
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Coincidentally I have been sent an actual interview today with the American neurologist David Eagleman about his new book "Icognito".
Basically he discusses the conscious ego as not center of personality, but practically periphery. Displaying us as much less self-governed than we dared to think even after the cancel of free will.
And accidentally, as I was about the relevance of time, time plays a major role in his thesis. Not only with its relativity ( hypothetical aliens of other time perception experiencing us slow like trees, etc.), but with its subjective stretching and shrinking in dependence of the psyche. ( Slow-motion when in surprise / accident ...)
Thus, the inconsitent sensation of time leaving us more arbitrary and unpredictabel than we used to suspect already.
-
Another point, though not connected to the whereabouts of mathematics as either invention or precondition, but relevant and very interesting still: Eagleman´s description of the ego as conflict between insight and impulse. With impulse often coming out superiour. ( And when regularly = criminal behaviour.)
Basically providing impulse as the background of evil, while in the same time developing effective methods for training of impulse suppression.
If this be true, it will present sensational means in pedagogy and disciplining.
And such a simple one!
Should be really great news.
Ruphus
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Date Nov. 22 2013 11:51:54
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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quote:
ORIGINAL: BarkellWH
Nevertheless, I think mathematical structures transcend, and are independent of, the human mind (forget Platonic Forms). When mathematicians make a "discovery," or solve a problem (Fermat's Last Theorem), they are bringing forth something that existed prior to their "discovery"; something that was just waiting for a human mind (or minds) to reach the level of sophistication required to "crack the code," so to speak.
Penrose identifies himself as a Platonist, but if you read on, it is in a very limited sense. He attributes the Platonic existence of the truth of Fermat's last theorem to its objectivity. He defines "objective" as "independent of cultural values and individual opinion". So far I'm with Penrose, and you seem to be as well. But under this definition "objective" is not the same thing as "eternal" or even "pre-existing".
A while ago it was raining hard here in Austin. While it was raining hard, "It is raining" was an objective truth, but it was true only temporarily, and people would disagree when it actually started and stopped.
Did Fermat's Last Theorem even exist before humans invented multiplication, let alone its shorthand of raising numbers to a power? (Fermat's Last Theorem is that if A, B and C are natural numbers, then there is no natural number N>2 such that A^N + B^N = C^N).
Many ethologists exhibit experiments that are interpreted to show their animal subjects can "count" up to fairly small numbers. Anthropologists tell us that not all adult humans can get beyond 3. I don't know of any evidence that animals can multiply.
Do snails sum infinite series and perform multiplications when they grow their shells in logarithmic spirals? I don't think so, any more than I believe a major league hitter starts solving differential equations when he sees the pattern made by the spinning red seams of the ball against the white cover, when the ball leaves the pitcher's hand.
When, even approximately, did Fermat's Last Theorem come into existence? More to the point, what difference would it make if we knew?
In fact I am genuinely curious to know why you believe mathematical structures (all mathematical structures?) transcend the human mind.
RNJ
Ahhh! Bach's Italian Concerto on the radio….there's a structure of consummate beauty which deserves to be eternal--or at least to last as long as the human race. Even played in equal-tempered tuning on the piano, not in the well-tempered tuning in which Bach conceived it.
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Date Nov. 22 2013 21:52:07
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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quote:
ORIGINAL: BarkellWH
quote:
So far I'm with Penrose, and you seem to be as well. But under this definition "objective" is not the same thing as "eternal" or even "pre-existing"....In fact I am genuinely curious to know why you believe mathematical structures (all mathematical structures?) transcend the human mind.
Because, as I stated previously, I think they exist as what I can only call "symmetries" in the natural order of the universe. There are certain "symmetries" that exist independently of the human mind. Mathematical structures are among those universal "symmetries" that do not require a human mind's perception to bring them into existence. They exist whether or not a human mind perceives them.
Cheers,
Bill
The detailed solutions of the Standard Model are so complex that much of what we know about subatomic physics comes from studying the mathematical symmetries of the theory, and the breaking of some symmetries. But the Standard Model is so far from intuitive experience that I think most people would grant that it's a human construction, and so are the mathematical symmetries employed to draw conclusions from it.
Of course, our physical theories are constrained by the working of the universe. They have to get close enough to the right answer enough of the time to be useful. But I don't think our physical theories are fully determined by the working of the universe. I don't think our mathematical structures are fully determined by the working of the universe either.
Intuitionist mathematics, Nonstandard Analysis and the usual "standard" mathematics taught in universities these days all get the same answer when applied to physical theories, but they are quite different logical structures. Intuitionist mathematics contains fewer mathematical objects than "standard mathematics", and is logically incompatible with it. "Standard mathematics" does not contain the infinitesimals of Nonstandard Analysis. Computations which are valid in Nonstandard Analysis result in contradictions in "Standard mathematics". Yet all three yield the same rational approximations to the solution of physical problems.
Matrix mechanics was the first logically consistent theory of quantum mechanics, yet you don't hear much about it in first year quantum mechanics courses, except maybe as a footnote.
Is there a more detailed description of the "universal symmetries" you speak of, and how they determine our mathematical structures?
I'm not just pulling your leg.
RNJ
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Date Nov. 23 2013 0:05:56
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Richard Jernigan)
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quote:
Is there a more detailed description of the "universal symmetries" you speak of, and how they determine our mathematical structures?
My entire academic and professional career has encompassed history, foreign affairs, and national security. I am neither a mathematician nor a physicist, although I have a strong layman's interest in those fields. My belief that certain "symmetries" exist, and that they include mathematical structures independent of the human mind's perception of those structures, is based on a careful reading of tracts by mathematicians (mentioned in my previous posts in this thread) and an intuitive sense that these mathematical structures have a reality in their own right.
I cannot offer a "proof" of my position, any more than anyone can offer a "proof" that such mathematical structures do not exist as a separate reality, independent of the human mind. Nevertheless, these mathematical structures possess an elegance that matches that of other symmetries that exist in the natural world, whether or not they are perceived by a human mind. Thus, intuitively, I see no reason why mathematical structures cannot as well.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 23 2013 1:32:13
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to guitarbuddha)
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quote:
This is the hub of the matter. You are, whether you admit it or not, substituting mathematics for the religion of your youth. You are using the same language (beauty and mystery), invoking the same arguments (insisting on parity of positive and negative proof). The first time I pointed this out you claimed it was not your intention. And maybe it isn't your conscious intention. But your need is clear.
The elegance of symmetries found in the natural world, of which I think mathematical structures to be one, has nothing whatsoever to do with religion. I use the terms "beauty" and "mystery" to describe that elegance, not as a substitute for the "religion of my youth"; rather, to describe a phenomenon that indeed (in my opinion) possesses elegance but is difficult to "prove" or explain how it came into being. Again, I emphasize that I have for the most part intuitively come to this conclusion (albeit after delving into the works of mathematicians and philosophers from Penrose to Russell), and I fully realize that it can be (and has been) challenged by others. I can live with that and sleep well at night!
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 23 2013 13:03:45
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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quote:
ORIGINAL: BarkellWH
I cannot offer a "proof" of my position, any more than anyone can offer a "proof" that such mathematical structures do not exist as a separate reality, independent of the human mind.
We agree.
quote:
Nevertheless, these mathematical structures possess an elegance that matches that of other symmetries that exist in the natural world, whether or not they are perceived by a human mind. Thus, intuitively, I see no reason why mathematical structures cannot as well.
Cheers,
Bill
At least one place where I believe we differ is that I have spent a fair amount of time in the company of mathematicians, some of them distinguished, and in doing mathematics myself.
Mathematicians are divided on the issue of Platonism, in part because they define it in different ways. Penrose, whom you cite as a Platonist, turns out to have a definition of Platonic existence that to me is trivial, and fails to represent the spirit of Platonism as I have encountered it among friends and acquaintances.
I have read a fair amount about the history of mathematics, the centuries of groping toward the concepts we use now. Sometimes the road forks, and produces two or more logically valid and practically applicable concepts addressing the same issues.
For example the Riemann integral and the Lebesgue integral in standard analysis both address the issues of defining and calculating length, area, volume and their generalizations. The extensions of these two structures to the Riemann-Stieltjes and Lebesgue-Stieltjes integrals extend their applicability. In the latter case, applicability extends in probability theory to integrals over spaces composed of events, not Euclidian spaces. Some of my friends have written papers on further generalizations of these concepts, published in respectable journals as original research.
The extensions make the proof of some of the standard theorems of elementary calculus more general. They put probability theory on a sound logical basis. This is the kind of thing that disposes me to see mathematics as a human activity.
I remain agnostic about the non-human existence of mathematics, since it is difficult, if not impossible for me to conceive what that might be.
Of course, mathematics has been one of our most productive tools in understanding the universe. But just as someone made the elegant set of chrome-vanadium steel wrenches my father used to work on his airplanes and automobiles, I think humans made the mathematical tools we use to understand and modify our environment.
Thanks for an engaging discussion.
RNJ
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Date Nov. 23 2013 18:54:13
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